pendula.nb 1
Numerical study of coupled oscillations, with N=3 degrees of
freedom.
For Ph106a, fall term 2001, Caltech.
To see the eigenfrequencies, I need to Fourier transform some qi t . In practice it s convenient to do a discrete Fourier
transform, but the price is that when I then ListPlot the answer, the frequency is in funny units which depend on the length
of the time interval over which qi t was evaluated.
Another oddity of the numerics is that I won t see delta functions in frequency space, just because I m only looking over a
finite interval of time. So I should see sharp peaks instead.
Click on the square brackets on the right hand side of the window to look at the contents of each section.
Simple harmonic motion, one oscillator
soln
NDSolve q  t q t 0,q 0 0,q 0 1 ,q, t,0,500 ,MaxSteps Infinity 1
q InterpolatingFunction 0.,500. ,
range soln 1,2,1,1 ;
Plot q t .soln, t,range 1 ,range 2 ,PlotPoints 10000,PlotRange 2,2
2
1.5
1
0.5
pendula.nb 2
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NumPts 2^18;
ValueTable
Table q t .soln, t,range 1 ,range 2 , range 2 range 1 NumPts ;
FourierTable Fourier ValueTable ;
pendula.nb 3
OneOscillator ListPlot Abs FourierTable ,
PlotRange 50,180 , 0,Apply Max,Abs FourierTable 1 ,PlotJoined True
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Coupled harmonic pendula, moderately weak coupling
ClearAll kappa
CoupledPendula q1  t q1 t kappa q1 t q2 t 0,
q2  t q2 t kappa 2 q2 t q1 t q3 t 0,
q3  t q3 t kappa q3 t q2 t 0,q1 0 0,
q2 0 0,q3 0 0,q1 0 1,q2 0 0,q3 0 0 ;
kappa 0.1
0.1
soln NDSolve CoupledPendula, q1,q2,q3 , t,0,500 ,MaxSteps Infinity 1
q1 InterpolatingFunction 0.,500. , ,
q2 InterpolatingFunction 0.,500. , ,
q3 InterpolatingFunction 0.,500. ,
range soln 1,2,1,1 ;
Plot q1 t .soln, t,range 1 ,range 2
1
pendula.nb 4
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100
-0.5
-1
Graphics
NumPts 2^18;
ValueTable
Table q1 t .soln, t,range 1 ,range 2 , range 2 range 1 NumPts ;
FourierTable Fourier ValueTable ;
pendula.nb 5
WeakCoupling ListPlot Abs FourierTable ,
PlotRange 50,180 , 0,Apply Max,Abs FourierTable 1 ,PlotJoined True
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Graphics
Coupled harmonic pendula, moderately strong coupling
ClearAll kappa
CoupledPendula q1  t q1 t kappa q1 t q2 t 0,
q2  t q2 t kappa 2 q2 t q1 t q3 t 0,
q3  t q3 t kappa q3 t q2 t 0,q1 0 0,
q2 0 0,q3 0 0,q1 0 1,q2 0 0,q3 0 0 ;
kappa 1
1
soln NDSolve CoupledPendula, q1,q2,q3 , t,0,500 ,MaxSteps Infinity 1
q1 InterpolatingFunction 0.,500. , ,
q2 InterpolatingFunction 0.,500. , ,
q3 InterpolatingFunction 0.,500. ,
range soln 1,2,1,1 ;
Plot q1 t .soln, t,range 1 ,range 2
1
0.75
pendula.nb 6
0.5
0.25
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-0.75
Graphics
NumPts 2^18;
ValueTable
Table q1 t .soln, t,range 1 ,range 2 , range 2 range 1 NumPts ;
FourierTable Fourier ValueTable ;
pendula.nb 7
StrongCoupling ListPlot Abs FourierTable ,
PlotRange 50,180 , 0,Apply Max,Abs FourierTable 1 ,PlotJoined True
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Graphics
Anharmonic pendula, moderately strong coupling
ClearAll kappa
CoupledPendula q1  t q1 t kappa q1 t q2 t 0,
q2  t q2 t ^3 kappa 2 q2 t q1 t q3 t 0,
q3  t q3 t kappa q3 t q2 t 0,q1 0 0,
q2 0 0,q3 0 0,q1 0 1,q2 0 0,q3 0 0 ;
kappa 1
1
soln NDSolve CoupledPendula, q1,q2,q3 , t,0,500 ,MaxSteps Infinity 1
q1 InterpolatingFunction 0.,500. , ,
q2 InterpolatingFunction 0.,500. , ,
q3 InterpolatingFunction 0.,500. ,
range soln 1,2,1,1 ;
Plot q1 t .soln, t,range 1 ,range 2
1
pendula.nb 8
0.5
100
-0.5
-1
Graphics
NumPts 2^18;
ValueTable
Table q1 t .soln, t,range 1 ,range 2 , range 2 range 1 NumPts ;
FourierTable Fourier ValueTable ;
pendula.nb 9
Anharmonic ListPlot Abs FourierTable ,
PlotRange 50,180 , 0,Apply Max,Abs FourierTable 1 ,PlotJoined True
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Graphics
Anharmonic pendula, moderately strong coupling, bigger amplitude
ClearAll kappa
CoupledPendula q1  t q1 t kappa q1 t q2 t 0,
q2  t q2 t ^3 kappa 2 q2 t q1 t q3 t 0,
q3  t q3 t kappa q3 t q2 t 0,q1 0 0,
q2 0 0,q3 0 0,q1 0 10,q2 0 0,q3 0 0 ;
kappa 1
1
soln NDSolve CoupledPendula, q1,q2,q3 , t,0,500 ,MaxSteps Infinity 1
q1 InterpolatingFunction 0.,500. , ,
q2 InterpolatingFunction 0.,500. , ,
q3 InterpolatingFunction 0.,500. ,
range soln 1,2,1,1 ;
Plot q1 t .soln, t,range 1 ,range 2
10
pendula.nb 10
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-10
Graphics
NumPts 2^18;
ValueTable
Table q1 t .soln, t,range 1 ,range 2 , range 2 range 1 NumPts ;
FourierTable Fourier ValueTable ;
pendula.nb 11
BigAnharmonic ListPlot Abs FourierTable ,
PlotRange 50,180 , 0,Apply Max,Abs FourierTable 1 ,PlotJoined True
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Show all Fourier plots
A single harmonic oscillator:
Show OneOscillator
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Three coupled pendula, where the springs between the pendula make only a weak perturbation on the period of the free
motion:
pendula.nb 12
Show WeakCoupling
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Graphics
Stronger coupling between the pendula splits the eigenfrequencies more:
Show StrongCoupling
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Just for fun: the MIDDLE pendulum is anharmonic, but we re watching the left hand one. Startlingly similar to the Strong-
Coupling case.
Show Anharmonic
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pendula.nb 13
But now we multiplied the amplitude by 10, and we get QUITE a different spectrum... Pretty weird that I only get two
peaks now---is
this an accident?
Show BigAnharmonic
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